The paper explores the connection between self-acceleration and the presence of ghosts in theories that modify gravity at short distances. The authors focus on a class of theories that, at distances shorter than the Hubble scale, reduce to a four-dimensional effective Lagrangian involving a relativistic scalar field \(\pi\), which is universally coupled to matter and has peculiar derivative self-interactions. They argue that for locally modified gravity theories, their generalization is the only one that allows for a robust implementation of the Vainshtein effect, which decouples the scalar from matter in gravitationally bound systems, necessary to recover agreement with solar system tests. The generalization involves an internal "Galilean" invariance, under which \(\pi\)'s gradient shifts by a constant. This symmetry constraints the structure of the \(\pi\) Lagrangian, leading to five terms that can yield significant nonlinearities without introducing ghosts. The authors show that these theories admit "self-accelerating" de Sitter solutions with no ghost-like instabilities and can support spherically symmetric, Vainshtein-like non-linear perturbations that are stable against small fluctuations. They also investigate the possible infrared completion of these theories at scales larger than the Hubble horizon, noting some theoretical and phenomenological challenges, such as superluminal excitations and the extreme sub-luminality of other excitations.The paper explores the connection between self-acceleration and the presence of ghosts in theories that modify gravity at short distances. The authors focus on a class of theories that, at distances shorter than the Hubble scale, reduce to a four-dimensional effective Lagrangian involving a relativistic scalar field \(\pi\), which is universally coupled to matter and has peculiar derivative self-interactions. They argue that for locally modified gravity theories, their generalization is the only one that allows for a robust implementation of the Vainshtein effect, which decouples the scalar from matter in gravitationally bound systems, necessary to recover agreement with solar system tests. The generalization involves an internal "Galilean" invariance, under which \(\pi\)'s gradient shifts by a constant. This symmetry constraints the structure of the \(\pi\) Lagrangian, leading to five terms that can yield significant nonlinearities without introducing ghosts. The authors show that these theories admit "self-accelerating" de Sitter solutions with no ghost-like instabilities and can support spherically symmetric, Vainshtein-like non-linear perturbations that are stable against small fluctuations. They also investigate the possible infrared completion of these theories at scales larger than the Hubble horizon, noting some theoretical and phenomenological challenges, such as superluminal excitations and the extreme sub-luminality of other excitations.